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This is a partial listing of the more
popular theorems, postulates and
properties
needed when working with Euclidean
proofs.
You need to have a thorough understanding of these items.
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Your textbook (and your teacher) may want you to
remember these theorems with slightly different wording.
Be sure to follow the directions from your teacher. |
The
"I need to know, now!"
entries are highlighted in blue.
General:
|
Reflexive Property |
A quantity is congruent (equal) to itself. a = a |
|
Symmetric Property |
If a = b,
then b = a. |
|
Transitive Property |
If a = b and
b = c, then a = c. |
| Addition Postulate |
If equal quantities are added to
equal quantities, the sums are equal. |
| Subtraction Postulate
|
If equal quantities are subtracted
from equal quantities, the differences are equal. |
| Multiplication
Postulate |
If equal quantities are multiplied
by equal quantities, the products are equal. (also Doubles of
equal quantities are equal.) |
| Division Postulate |
If equal quantities are divided by
equal nonzero quantities, the quotients are equal. (also Halves of
equal quantities are equal.) |
| Substitution
Postulate |
A quantity may be substituted for
its equal in any expression. |
| Partition Postulate |
The whole is equal to the sum of its parts.
Also: Betweeness of Points: AB + BC = AC
Angle Addition Postulate: m<ABC + m<CBD = m<ABD |
| Construction |
Two points determine a straight
line.
|
| Construction |
From a given point on (or not on)
a line, one and only one perpendicular can be drawn to the line. |
Angles:
| Right Angles |
All right angles are congruent.
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| Straight Angles |
All straight angles
are congruent.
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| Congruent Supplements |
Supplements of the same angle, or
congruent angles, are congruent. |
| Congruent Complements |
Complements of the same angle, or
congruent angles, are congruent. |
| Linear Pair |
If two angles form a
linear pair, they are supplementary.
|
| Vertical Angles |
Vertical angles are congruent.
|
| Triangle Sum |
The sum of the interior angles of a triangle is
180º.
|
| Exterior Angle |
The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two non-adjacent interior
angles.
The measure of an exterior angle of a triangle is greater than
either non-adjacent interior angle. |
Base Angle Theorem
(Isosceles Triangle) |
If two sides of a triangle are
congruent, the angles opposite these sides are congruent. |
Base Angle Converse
(Isosceles Triangle) |
If two angles of a triangle are
congruent, the sides opposite these angles are congruent. |
Triangles:
| Side-Side-Side (SSS) Congruence |
If three sides of one triangle are congruent to
three sides of another triangle, then the triangles are
congruent. |
| Side-Angle-Side (SAS) Congruence |
If two sides and the included angle of one triangle
are congruent to the corresponding parts of another triangle, the
triangles are congruent. |
| Angle-Side-Angle (ASA) Congruence |
If two angles and the included side of one triangle
are congruent to the corresponding parts of another triangle, the
triangles are congruent. |
| Angle-Angle-Side (AAS) Congruence |
If two angles and the non-included side
of one triangle are congruent to the corresponding parts of another
triangle, the triangles are congruent. |
| Hypotenuse-Leg (HL) Congruence (right triangle) |
If the hypotenuse and leg of one right
triangle are congruent to the corresponding parts of another right
triangle, the two right triangles are congruent. |
|
CPCTC |
Corresponding parts
of congruent triangles are congruent. |
| Angle-Angle (AA) Similarity |
If two angles of one triangle are congruent to two
angles of another triangle, the triangles are
similar. |
|
SSS for Similarity |
If the three sets of
corresponding sides of two triangles are in proportion, the
triangles are similar. |
|
SAS for Similarity |
If an angle of one
triangle is congruent to the corresponding angle of another triangle
and the lengths of the sides including these angles are in
proportion, the triangles are similar. |
| Side Proportionality |
If two triangles are
similar, the
corresponding sides are in proportion. |
Mid-segment Theorem
(also called mid-line) |
The segment connecting the midpoints of
two sides of a triangle is parallel
to the third side and is half as
long. |
| Sum of Two Sides |
The sum of the
lengths of any two sides of a triangle must be greater than the
third side
|
| Longest Side |
In a triangle, the longest side is
across from the largest angle.
In a triangle, the largest angle is across from the longest side. |
| Altitude Rule |
The altitude
to the hypotenuse of a right triangle is the mean proportional
between the segments into which it divides the hypotenuse. |
| Leg Rule |
Each leg
of a right triangle is the mean proportional between the hypotenuse
and the projection of the leg on the hypotenuse. |
Parallels:
| Corresponding Angles
|
If two parallel lines are cut by a transversal, then
the pairs of corresponding angles are congruent. |
| Corresponding Angles Converse
|
If two lines are cut by a transversal and the
corresponding angles are congruent, the lines are
parallel. |
Alternate Interior Angles
|
If two
parallel lines
are cut by a transversal, then the alternate
interior angles are congruent. |
|
Alternate Exterior Angles |
If two
parallel lines are cut by a
transversal, then the alternate exterior angles
are congruent. |
|
Interiors on Same Side
|
If two
parallel lines
are cut by a transversal, the interior angles on
the same side of the transversal are
supplementary. |
Alternate Interior Angles
Converse |
If two lines
are cut by a transversal and the alternate
interior angles are congruent, the lines are
parallel. |
Alternate Exterior Angles
Converse |
If two lines
are cut by a transversal and the alternate
exterior angles are congruent, the lines are
parallel. |
| Interiors on Same
Side Converse |
If two lines are cut by a
transversal and the interior angles on the same
side of the transversal are supplementary, the
lines are parallel. |
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Quadrilaterals:
|
Parallelograms
|
About Sides
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* If a quadrilateral
is a parallelogram, the opposite
sides are parallel.
* If a
quadrilateral is a parallelogram, the opposite
sides are congruent. |
|
About Angles |
* If a quadrilateral
is a parallelogram, the opposite
angles are congruent.
* If a
quadrilateral is a parallelogram, the
consecutive angles are supplementary. |
|
About Diagonals |
* If a quadrilateral
is a parallelogram, the diagonals
bisect each other.
* If a
quadrilateral is a parallelogram, the diagonals
form two congruent triangles. |
|
Parallelogram Converses
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About Sides
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* If both pairs of
opposite sides of a quadrilateral
are parallel, the quadrilateral is a parallelogram.
* If both pairs of
opposite sides of a quadrilateral
are congruent, the quadrilateral is a
parallelogram. |
|
About Angles |
* If both pairs of
opposite angles of a quadrilateral
are congruent, the quadrilateral is a
parallelogram.
* If the
consecutive angles of a quadrilateral are
supplementary, the quadrilateral is a parallelogram. |
|
About Diagonals
|
* If the diagonals of
a quadrilateral bisect each
other, the quadrilateral is a
parallelogram.
* If the diagonals
of a quadrilateral form two
congruent triangles, the quadrilateral is a
parallelogram. |
| Parallelogram |
If
one pair of sides of a quadrilateral is BOTH parallel and
congruent, the quadrilateral is a parallelogram. |
|
Rectangle |
If a
parallelogram has one right angle it is a rectangle |
| A
parallelogram is a rectangle if and only if its diagonals are
congruent. |
| A
rectangle is a parallelogram with four right angles. |
|
Rhombus |
A
rhombus is a parallelogram with four congruent sides. |
|
If a parallelogram has two consecutive sides congruent, it is a
rhombus. |
| A
parallelogram is a rhombus if and only if each diagonal bisects
a pair of opposite angles. |
| A
parallelogram is a rhombus if and only if the diagonals are
perpendicular. |
|
Square |
A
square is a parallelogram with four congruent sides and four
right angles. |
| A
quadrilateral is a square if and only if it is a rhombus and a
rectangle. |
| Trapezoid |
A
trapezoid is a quadrilateral with exactly one pair of parallel
sides. |
|
Isosceles Trapezoid |
An
isosceles trapezoid is a trapezoid with congruent legs. |
| A
trapezoid is isosceles if and only if the base angles are
congruent |
| A
trapezoid is isosceles if and only if the diagonals are
congruent |
| If a
trapezoid is isosceles, the opposite angles are supplementary. |
Circles:
| Radius |
In a circle, a radius
perpendicular to a chord bisects the chord and the arc. |
| In a circle, a radius that bisects
a chord is perpendicular to the chord. |
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In a circle, the perpendicular bisector of a chord
passes through the center of the circle.
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| If a line is tangent to a circle,
it is perpendicular to the radius drawn to the point of tangency. |
| Chords |
In a circle, or congruent circles, congruent
chords are equidistant from the center. (and converse)
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| In a circle, or congruent circles,
congruent chords have congruent arcs. (and converse0 |
| In a circle, parallel chords
intercept congruent arcs |
| In the same circle, or congruent
circles, congruent central angles have congruent chords (and
converse) |
| Tangents |
Tangent segments to a circle from
the same external point are congruent |
| Arcs |
In the same circle, or congruent
circles, congruent central angles have congruent arcs. (and
converse) |
| Angles |
An angle inscribed in a
semi-circle is a right angle. |
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In a circle, inscribed circles that intercept the
same arc are congruent. |
| The opposite angles in a cyclic
quadrilateral are supplementary |
| In a circle, or congruent circles,
congruent central angles have congruent arcs. |
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